Given two simplicial topological spaces $X_{\bullet}$ and $Y_{\bullet}$ (i.e. a simplicial object in Top) and a continuous map between their geometric realizations $f \colon \lvert X_{\bullet} \rvert \to \lvert Y_{\bullet} \rvert$. Is $f$ homotopic to $\lvert \varphi_{\bullet} \rvert$ for a map $\varphi_{\bullet}$ of simplicial spaces?
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asked Mar 7, 2012 at 17:02
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$\begingroup$ To parallel ordinary simplicial approximation, shouldn't you allow $X$ to be replaced with some notion of its 'subdivision'? $\endgroup$– Jeff StromCommented Mar 7, 2012 at 19:29
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$\begingroup$ @Jeff: That's true, but I had no idea what that should be in case of simplicial spaces. $\endgroup$– Ulrich PennigCommented Mar 7, 2012 at 19:58
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$\begingroup$ The motivation for this admittedly naive question was the following setup: Consider a topological category C and homotopy classes of maps from a topological space X to BC. Now replace X by the geometric realization Y of the nerve of a good cover. This yields two simplicial spaces and I would understand [Y,BC] much better, if I had some kind of theorem like the above. $\endgroup$– Ulrich PennigCommented Mar 7, 2012 at 20:03
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2 Answers
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The answer is no. For an arbitrary simplicial space $X_\bullet$ we can consider $|X_\bullet|$ as a constant simplicial space, lets call this $Y_\bullet$. Then there is clearly the identity $|X_\bullet| \to |Y_\bullet|$, but there is in general no nontrivial map $X_\bullet \to Y_\bullet$ (take e.g. BG for a top. group $G$).
answered Mar 7, 2012 at 17:22
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I'm just reformulating your question in simplicial case. If you consider the category of simplicial sets $\mathbf{sSet}$ you can formulate your question as follows: Is the diagonal functor $diag: [\mathbf{\Delta}^{op},\mathbf{sSet}]\rightarrow \mathbf{sSet}$ from bisimplicial sets to simplicial sets homotopicaly full ? The diagonal functor is actually the realization functor. If we put the diagonal model structure on $ [\mathbf{\Delta}^{op},\mathbf{sSet}]$ then $diag$ is a Quillen equivalence. As a consequence, if $f: diag(X_{\bullet, \bullet})\rightarrow diag(Y_{\bullet, \bullet})$ is a bisimplicial map and $X_{\bullet, \bullet}$, $Y_{\bullet, \bullet}$ are fibrant-cofibrant bisimplicial sets (in the diagonal model structure), then there exists $g: X_{\bullet, \bullet}\rightarrow Y_{\bullet, \bullet}$ such that $diag(g)$ is homotopic to $f$.
